When the early muslim community was being persecuted by the Meccans, a faction escaped to Ethiopia and took refuge with its king, and when the Meccans heard about this, they sent a delegation asking him to give the muslims up to them, to which he refused, and having sent the delegation away, granted amnesty to the muslim refugees and allowed them to continue staying as his guests.

So for the early muslim Caliphates, the Ethiopian monarchy's positive role in the formative years of Islam gave it an informal special status. And to my knowledge, there were no expeditions mounted against them throughout the Rashidun, Umayyad, or Abbasid Caliphates.

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Just wanted to say that Rome didn't really reach far enough into Africa to reach Ethiopia, or its dominant power at the time, Axum, aside from some trade relations, and that Byzantium in its early days before the Muslim invasions had extensive diplomatic and trade relations with Axum, including a joint operation into Southern Yemen against a genocidal Jewish King. Can't really speak to the others, but regardless of its desirability, or rather lack thereof for the Romans in terms of conquest, it wasn't really practical, and they were allies of Byzantium.

The Ottomans did tried to attack Ethiopia. An Emir named Ahmed Grañ assembled a major force and attacked Ethiopia with Ottoman support in exchange for suzerainty. He's very successful at first, taking and looting the stronghold of Amba Geshen in 1538, where the accumulated wealth of the kings of Ethiopia had been held untouched for centuries. The Ethiopian king, Lebna Dengel then left as a fugitive and died shortly, but have had requested Portuguese assistance for driving out Ahmed. Eventually, Ahmed faced combined Ethiopian-Portuguese forces happened in battle at Wofla on 1542, where Ahmed successfully defeated the combined forces and taking numerous of Portuguese captives. However, things started to break apart here. The Ottomans wanted to use these captives as bargaining chips to negotiate with the Portuguese and demanded that they be handed over unharmed to the custody of Ottoman officials in Yemen. Ahmed refused this and executed most of the captives. The Ottomans were enraged and abandoned Ahmed taking with them the remaining Portuguese captives that are still alive. Without Ottoman support, Ahmed can't capitalize his victory and thus giving the Ethiopians a chance to regroup. A year later, The Ethiopians counterattacked at the battle of Wayna Daga in which Ahmed was killed. The remaining armies of his regrouped under Ahmed's nephew Abbas. But still, without Ottoman support he can't face the Ethiopians and their territory were reduced to coastal lowlands around Zeyla